Table of contents

Under certain conditions on a function we prove a theorem asserting that the convergence almost everywhere of trigonometric Fourier series for all functions of class implies the convergence over cubes of the multiple Fourier series and all its conjugates for an arbitrary function , . It follows from this and an earlier result of the author on the convergence almost everywhere of Fourier series of functions of one variable and class that if , , then the Fourier series of  and all its conjugates converge over cubes almost everywhere.

We consider quasilinear elliptic non-diagonal systems of equations with strong non-linearity with respect to the gradient. We have already shown that the generalized solution of this problem is Hölder continuous in the neighbourhood of points of the domain at which the norm of the gradient of the solution is sufficiently small in the Morrey space . We estimate the Hölder norm of the solution in the neighbourhood of such points in terms of its norm in the Sobolev space . We obtain a similar result under the Dirichlet boundary condition for points situated in the neighbourhood of the boundary.

We refine the Fuchs inequalities obtained by Corel for systems of linear meromorphic differential equations given on the Riemann sphere. Fuchs inequalities enable one to estimate the sum of exponents of the system over all its singular points. We refine these well-known inequalities by considering the Jordan structure of the leading coefficient of the Laurent series for the matrix of the right-hand side of the system in the neighbourhood of a singular point.

We consider systems of equations of the form , where the are the Fourier transforms of distributions with fixed compact supports, and show that the average density of roots of such systems is determined by the geometry of the convex hulls of the supports of the distributions as their product in the ring of convex bodies.

Under certain natural restrictions, we give a complete solution of the interpolation problem in spaces of entire functions of exponential type whose conjugate diagram is contained in a given convex domain. We also give a complete solution of the problem of finding a fundamental principle for arbitrary non-trivial closed subspaces of functions analytic in a convex domain that are invariant under differentiation and admit spectral synthesis.

We study three-dimensional exceptional canonical hypersurface singularities which do not satisfy the condition of well-formedness. The result obtained completes the classification of three-dimensional exceptional log canonical hypersurface singularities begun in [4].

We study Fano-Mori contractions with fibres of dimension at most one satisfying the semistability assumption. In particular, we give a new proof of the existence of semistable 3-fold flips.

Multiple points of the spectrum in the reduction are separated by introducing a non-semisimple intermediate subalgebra and a weight scheme different from the Gel'fand-Tsetlin scheme. We suggest a method of constructing a weight basis in the space of a finite-dimensional irreducible representation of . The elements of this basis are labelled by such weight schemes. We also study the category of finite-dimensional highest-weight representations of this intermediate Lie algebra.

A theorem of Fine and Wilf expresses the interaction property of periods, which is a basic property of periodic words. An arbitrary word with given periods p and q also has a "derived" period gcd(p,q) if the length of the word is greater than some critical value called the length of interaction. In this paper we consider a similar property for arbitrary periodic partial words and give a sharp linear bound for the length of interaction.

We prove that conic bundles with sufficiently big discriminant locus cannot be birationally transformed into fibrations whose generic fibre has numerically trivial canonical divisor.